The Pentagon gives a name to many of its significant overseas “contingencies” — but not the current, open-ended one in Iraq.

Afghanistan is “Operation Enduring Freedom.” The Iraq war went from “Operation Iraqi Freedom” to “Operation New Dawn” as U.S. troops methodically withdrew. The mission to kill or capture Osama bin Laden was “Operation Neptune Spear.” The 1990s operation to aid the Kurds in northern Iraq was “Operation Provide Comfort.”

President Obama has ordered a substantial new military mission in Iraq that involves hundreds of U.S. advisers, humanitarian air drops, daily surveillance flights and nearly 100 airstrikes on the Islamic State terrorist army. Swarming over Iraq are spy planes, armed drones, and Air Force and Navy jet fighters.

But all of this action lacks a cohesive umbrella — a name.

“There is no legal significance in the naming of operations, nor does it affect legal authority,” a Defense Department spokeswoman said. “No official name has been given to the series of targeted strikes we have taken, and we do not expect a name to be assigned.”

The spokeswoman said the administration has complied with the 1973 War Powers Resolution by notifying Congress in four separate reports: an assessment of security at the U.S. Embassy in Baghdad, the insertion of hundreds of U.S. military advisers, the expansion of that team, and humanitarian relief and airstrikes.

And the Pentagon says it will not be awarding campaign ribbons for the Iraq operation.

Insiders say the no-name mission may stem from Mr. Obama’s desire to keep the Iraq war in the past.

One Pentagon bureaucrat said the administration does not want to send the message that “we are officially re-engaged in Iraq.”

Mr. Obama withdrew all forces in December 2011 after having campaigned on ending a war started by President George W. Bush. He has said he was elected to end wars, not start them. Naming the current operation would more firmly link him historically to what is a limited war against the al Qaeda-spinoff Islamic State.

Not all foreign insertions of U.S. troops get a name. For example, the Pentagon last year sent military advisers to Uganda to help track down Joseph Kony, the leader of the brutal Lord’s Resistance Army. It received no name.